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Single-mode guiding properties of subwavelength-diameter silica and silicon wire waveguides

Limin Tong, Jingyi Lou, and Eric Mazur

Open Access Open Access

Abstract

Single-mode optical wave guiding properties of silica and silicon subwavelength-diameter wires are studied with exact solutions of Maxwell’s equations. Single mode conditions, modal fields, power distribution, group velocities and waveguide dispersions are studied. It shows that air-clad subwavelength-diameter wires have interesting properties such as tight-confinement ability, enhanced evanescent fields and large waveguide dispersions that are very promising for developing future microphotonic devices with subwavelength-width structures.

©2004 Optical Society of America

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Figures (12)
Fig. 1.
Fig. 1. Mathematic model of an air-clad cylindrical wire waveguide.

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Fig. 2.
Fig. 2. Numerical solutions of propagation constant (β) of air-clad silica wire at 633-nm wavelength. Solid line, fundamental mode. Dotted lines, high-order modes. Dashed line, critical diameter for single-mode operation (DSM ).

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Fig. 3.
Fig. 3. Numerical solutions of propagation constant (β) of air-clad silicon wire at 1.5-µm wavelength. Solid line, fundamental mode. Dotted lines, high-order modes. Dashed line, critical diameter for single-mode operation (DSM ).

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Fig. 4.
Fig. 4. Single mode condition of an air-clad silica and silicon wires. Solid line, critical diameter for single-mode operation. Dotted line, wavelength in media.

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Fig. 5.
Fig. 5. Electric components of HE11 modes of silica wires at 633-nm wavelength with different diameters in cylindrical coordination. Normalizations are applied as: εer(r=0)=1 and eΦ(r=0)=1. Wire diameters are arrowed to each curve in unit of nm.

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Fig. 6.
Fig. 6. Z-direction Poynting vectors of silica wires at 633-nm wavelength with diameters of (A) 400 nm and (B) 200 nm. Mesh, field inside the core. Gradient, field outside the core.

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Fig. 7.
Fig. 7. Fractional power of the fundamental modes inside the core of (A) silica wire at 633-nm wavelength, (B) silica wire at 1.5-µm wavelength and (C) silicon wire at 1.5-µm wavelength. Dashed line, critical diameter for single mode operation.

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Fig. 8.
Fig. 8. Effective diameters of the light fields of the fundamental modes. Solid line, Deff. Dotted line, real diameter. Dashed line, critical diameter for single mode operation. (A) silica wire at 633-nm wavelength, (B) silica wire at 1.5-µm wavelength and (C) silicon wire at 1.5-µm wavelength.

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Fig. 9.
Fig. 9. Diameter-dependent group velocities of the fundamental modes of air-clad (A) silica wire at 633- nm and 1.5-µm wavelengths and (B) silicon wire at 1.5-µm wavelength.

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Fig. 10.
Fig. 10. Wavelength-dependent group velocities of the fundamental modes of air-clad (A) lica wire and (B) silicon wire with different diameters (wire diameters are labeled on each rve in unit of nm)

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Fig. 11.
Fig. 11. Diameter-dependent waveguide dispersion of fundamental modes of air-clad (A) silica wire at 633-nm and 1.5-µm wavelengths and (B) silicon wire at 1.5-µm wavelengths.

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Fig. 12.
Fig. 12. Wavelength-dependent waveguide dispersion of fundamental modes of air-clad (A) silica wire and (B) silicon wire with different wire diameters (the wire diameter is labeled on each curve in unit of nm). Material dispersion is plotted in dotted line.

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Equations (23)

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n ( r ) = { n 1 , 0 < r < a , n 2 , a ≤ r < ∞
( ∇ 2 + n 2 k 2 − β 2 ) e ⇀ = 0 ,
( ∇ 2 + n 2 k 2 − β 2 ) h ⇀ = 0
{ J v ′ ( U ) UJ v ( U ) + K v ′ ( W ) WK v ( W ) } { J v ′ ( U ) UJ v ( U ) + n 2 2 K v ′ ( W ) n 1 2 WK v ( W ) } = ( v β kn 1 ) 2 ( V UW ) 4
J 1 ( U ) UJ 0 ( U ) + K 1 ( W ) WK 0 ( W ) = 0
n 1 2 J 1 ( U ) UJ 0 ( U ) + n 2 2 K 1 ( W ) WK 0 ( W ) = 0
n 2 − 1 = 0.6961663 λ 2 λ 2 − ( 0.0684043 ) 2 + 0.4079426 λ 2 λ 2 − ( 0.1162414 ) 2 + 0.8974794 λ 2 λ 2 − ( 9.896161 ) 2
n 2 = 11.6858 + 0.939816 λ 2 + 0.000993358 λ 2 − 1.22567
V = 2 π · a λ 0 · ( n 1 2 − n 2 2 ) 1 2 ≈ 2.405 .
{ J 1 ′ ( U ) UJ 1 ( U ) + K 1 ′ ( W ) WK 1 ( W ) } { J 1 ′ ( U ) UJ 1 ( U ) + n 2 2 K 1 ′ ( W ) n 1 2 WK 1 ( W ) } = ( β kn 1 ) 2 ( V UW ) 4
{ E ⇀ ( r , ϕ , z ) = ( e r r ̂ + e ϕ ϕ ̂ + e z z ̂ ) e i β z e − i ω t , H ⇀ ( r , ϕ , z ) = ( h r r ̂ + h ϕ ϕ ̂ + h z z ̂ ) e i β z e − i ω t
e r = − a 1 J 0 ( UR ) + a 2 J 2 ( UR ) J 1 ( U ) · f 1 ( ϕ ) ,
e ϕ = − a 1 J 0 ( UR ) + a 2 J 2 ( UR ) J 1 ( U ) · g 1 ( ϕ ) ,
e z = − iU a β J 1 ( UR ) J 1 ( U ) · f 1 ( ϕ )
e r = − U W a 1 K 0 ( WR ) − a 2 K 2 ( WR ) K 1 ( W ) · f 1 ( ϕ ) ,
e ϕ = − U W a 1 K 0 ( WR ) − a 2 K 2 ( WR ) K 1 ( W ) · g 1 ( ϕ ) ,
e z = − iU α β K 1 ( WR ) K 1 ( W ) · f 1 ( ϕ )
S z 1 = 1 2 ( ε 0 μ 0 ) 1 2 kn 1 2 β J 1 2 ( U ) [ a 1 a 3 J 0 2 ( UR ) + a 2 a 4 J 2 2 ( UR ) + 1 − F 1 F 2 2 J 0 ( UR ) J 2 ( UR ) cos ⁡ ( 2 ϕ ) ]
S z 2 = 1 2 ( ε 0 μ 0 ) 1 2 kn 1 2 β K 1 2 ( W ) U 2 W 2 [ a 1 a 5 K 0 2 ( WR ) + a 2 a 6 K 2 2 ( WR ) − 1 − 2 Δ − F 1 F 2 2 K 0 ( WR ) K 2 ( WR ) cos ⁡ ( 2 ϕ ) ]
η = ∫ 0 a S z 1 dA ∫ 0 a S z 1 dA + ∫ a ∞ S z 2 dA
{ ∫ 0 D eff S z 1 dA ∫ 0 a S z 1 dA + ∫ a ∞ S z 2 dA = 86.5 % , ( if D eff ≤ a ) , ∫ 0 a S z 1 dA + ∫ a D eff S z 1 dA ∫ 0 a S z 1 dA + ∫ a ∞ S z 2 dA = 86.5 % , ( if D eff > a ) .
v g = c n 1 2 · β k · 1 1 − 2 Δ ( 1 − η ) .
D w = d ( v g − 1 ) d λ .
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